Subgroups — #19146948

Order 5504.

388 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583604	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583550	C₂	1	1	yes	maximal, characteristic
3	#14583296	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583231	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583229	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583219	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583217	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583188	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583183	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583129	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583128	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582986	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×C₂²⋊D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
14	#106×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#106×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621898	—	2	1	no	—
17	C₂×#10621873	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₄₃	C₂²	1	1	yes	characteristic
19	#13063324	C₂²	1	1	yes	characteristic
20	#13063324	C₂²	1	1	yes	characteristic
21	#13063315	—	2	1	no	—
22	#13063310	—	2	1	no	—
23	#13063308	—	2	1	no	—
24	#13063305	C₂²	1	1	yes	characteristic
25	#13063304	—	2	1	no	—
26	#13063304	C₂²	1	1	yes	characteristic
27	#13063303	—	2	1	no	—
28	#13063303	—	2	1	no	—
29	#13063300	C₂²	1	1	yes	characteristic
30	#13063299	C₂²	1	1	yes	characteristic
31	#13063290	C₂²	1	1	yes	characteristic
32	#13063287	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₄₃	—	2	1	no	—
34	C₄×C₂²⋊D₄₃	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
37	#13063257	C₂²	1	1	yes	characteristic
38	#13063256	C₂²	1	1	yes	characteristic
39	#13063256	C₂²	1	1	yes	characteristic
40	#13063256	C₂²	1	1	yes	characteristic
41	#13063256	C₂²	1	1	yes	characteristic
42	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
43	#13063250	C₂²	1	1	yes	characteristic
44	#13063248	C₂²	1	1	yes	characteristic
45	#13063248	C₂²	1	1	yes	characteristic
46	#13063247	C₂²	1	1	yes	characteristic
47	#13063245	C₂²	1	1	yes	characteristic
48	#13063245	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
55	#106×C₄₃	C₂²	1	1	yes	characteristic
56	#106×C₄₃	C₂²	1	1	yes	characteristic
57	#106×C₄₃	C₂²	1	1	yes	characteristic
58	#106×C₄₃	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	—	2	1	no	—
60	C₂²×C₄×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₄₃	—	2	1	no	—
63	C₂×C₂²⋊D₄₃	D₄	1	1	yes	characteristic
64	#10621898	—	4	4	no	—
65	C₂×D₁₇₂	—	2	1	no	—
66	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₄₃	—	2	1	no	—
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	—	2	1	no	—
70	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
71	C₂×Dic₈₆	—	2	1	no	—
72	C₂×Dic₈₆	—	2	1	no	—
73	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₄₃	—	2	2	no	—
75	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₄₃	—	2	1	no	—
78	C₄⋊C₄×C₄₃	—	2	1	no	—
79	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₄₃	—	2	1	no	—
84	C₂²⋊Dic₄₃	D₄	1	1	yes	characteristic
85	(C₂×D₄₃)⋊C₄	—	2	1	no	—
86	(C₂×D₄₃)⋊C₄	—	2	1	no	—
87	(C₂×D₄₃)⋊C₄	—	2	1	no	—
88	(C₂×D₄₃)⋊C₄	—	2	1	no	—
89	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
90	C₄⋊Dic₄₃	—	2	1	no	—
91	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
93	Dic₄₃⋊C₄	—	2	1	no	—
94	Dic₄₃⋊C₄	—	2	1	no	—
95	Dic₄₃⋊C₄	—	2	1	no	—
96	Dic₄₃⋊C₄	—	2	1	no	—
97	Dic₄₃⋊C₄	—	2	1	no	—
98	Dic₄₃⋊C₄	—	2	1	no	—
99	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
102	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
104	#10621873	—	2	2	no	—
105	C₈×D₄₃	—	2	2	no	—
106	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₄₃	—	2	1	no	—
108	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	—	8	1	no	—
117	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₄₃	—	4	2	no	—
120	C₂²⋊D₄₃	SD₁₆	1	4	yes	complemented
121	C₂×Dic₄₃	—	2	1	no	—
122	C₂×Dic₄₃	—	2	1	no	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
127	D₁₇₂	—	2	1	no	—
128	D₁₇₂	—	2	1	no	—
129	D₁₇₂	—	4	2	no	—
130	C₄×D₄₃	—	2	1	no	—
131	C₄×D₄₃	—	2	1	no	—
132	C₄×D₄₃	—	4	2	no	—
133	C₄×D₄₃	—	4	2	no	—
134	C₄×D₄₃	—	8	1	no	—
135	Dic₈₆	—	2	1	no	—
136	Dic₈₆	—	2	1	no	—
137	Dic₈₆	—	4	2	no	—
138	Dic₈₆	—	8	1	no	—
139	C₈×C₄₃	—	2	1	no	—
140	C₈×C₄₃	#41	1	2	yes	complemented
141	C₄₃⋊C₈	—	2	1	no	—
142	C₄₃⋊C₈	—	2	1	no	—
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	C₂×SD₁₆	1	2	yes	—
146	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₄₃	—	2	1	no	—
148	C₂×D₄₃	—	2	1	no	—
149	C₂×D₄₃	—	4	1	no	—
150	C₂×D₄₃	—	4	1	no	—
151	C₂×D₄₃	—	4	1	no	—
152	C₂×D₄₃	C₂×SD₁₆	1	2	yes	—
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	—	4	1	no	—
155	C₄×C₄₃	—	4	1	no	—
156	C₄×C₄₃	—	4	1	no	—
157	C₄×C₄₃	—	4	1	no	—
158	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
159	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₄₃	—	2	1	no	—
161	Dic₄₃	—	2	1	no	—
162	Dic₄₃	—	4	1	no	—
163	Dic₄₃	—	4	1	no	—
164	Dic₄₃	—	4	1	no	—
165	Dic₄₃	C₂×SD₁₆	1	2	yes	—
166	#3335	—	43	1	no	maximal, Sylow
167	C₂×C₄₃	—	2	2	no	—
168	C₂×C₄₃	#547	1	1	yes	characteristic
169	C₂×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
170	C₂×C₄₃	#577	1	1	yes	characteristic
171	D₄₃	—	2	2	no	—
172	D₄₃	—	8	1	no	—
173	#554	—	43	1	no	—
174	#547	—	43	1	no	—
175	#498	—	43	1	no	—
176	#483	—	43	1	no	—
177	#483	—	43	1	no	—
178	#481	—	43	1	no	—
179	#481	—	43	1	no	—
180	#476	—	43	1	no	—
181	#475	—	43	1	no	—
182	#465	—	43	1	no	—
183	#465	—	43	1	no	—
184	C₄×SD₁₆	—	43	1	no	—
185	C₈×D₄	—	43	1	no	—
186	C₂×#106	—	43	1	no	—
187	C₂×#106	D₄₃	1	1	yes	characteristic, complemented
188	C₄₃	#3335	1	1	yes	characteristic, complemented, Sylow
189	C₂×#41	—	86	1	no	—
190	C₂×SD₁₆	—	43	1	no	—
191	C₂²×C₈	—	43	1	no	—
192	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
193	C₄⋊Q₈	—	43	1	no	—
194	C₄⋊Q₈	—	43	1	no	—
195	#125	—	86	1	no	—
196	#123	—	86	1	no	—
197	#123	—	86	1	no	—
198	C₂²⋊Q₈	—	43	1	no	—
199	C₂²⋊Q₈	—	43	1	no	—
200	C₂²⋊Q₈	—	86	1	no	—
201	C₂²⋊Q₈	—	86	1	no	—
202	C₂²⋊Q₈	—	86	1	no	—
203	#121	—	43	1	no	—
204	#121	—	43	1	no	—
205	C₄×Q₈	—	43	1	no	—
206	C₄×D₄	—	43	1	no	—
207	C₄×D₄	—	43	1	no	—
208	C₄×D₄	—	86	1	no	—
209	C₂×C₄⋊C₄	—	43	1	no	—
210	C₂×C₄⋊C₄	—	43	1	no	—
211	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
213	#106	—	43	1	no	—
214	#106	—	43	1	no	—
215	#106	—	43	1	no	—
216	#106	—	43	1	no	—
217	#106	—	43	1	no	—
218	#106	C₂×D₄₃	1	1	yes	characteristic
219	#106	C₂×D₄₃	1	1	yes	characteristic
220	#106	C₂×D₄₃	1	1	yes	characteristic
221	#106	C₂×D₄₃	1	1	yes	characteristic
222	C₄⋊C₈	—	43	1	no	—
223	#103	—	43	1	no	—
224	#103	—	43	1	no	—
225	#103	—	43	1	no	—
226	#102	—	43	1	no	—
227	#102	—	43	1	no	—
228	#102	—	43	1	no	—
229	C₂²⋊C₈	—	43	1	no	—
230	C₂²⋊C₈	—	43	1	no	—
231	C₄×C₈	—	43	1	no	—
232	#41	—	172	4	no	—
233	C₂×Q₈	—	43	1	no	—
234	C₂×Q₈	—	86	1	no	—
235	C₂×Q₈	—	86	1	no	—
236	C₂×D₄	—	43	1	no	—
237	C₂×D₄	—	43	1	no	—
238	C₂×D₄	—	86	1	no	—
239	C₂×D₄	—	86	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	43	1	no	—
243	C₂²×C₄	—	86	1	no	—
244	C₂²×C₄	—	86	1	no	—
245	C₂²×C₄	—	86	1	no	—
246	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
247	SD₁₆	—	86	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	43	1	no	—
250	C₂×C₈	—	43	1	no	—
251	C₂×C₈	—	86	2	no	—
252	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	43	1	no	—
257	C₄⋊C₄	—	43	1	no	—
258	C₄⋊C₄	—	43	1	no	—
259	C₄⋊C₄	—	86	1	no	—
260	C₄⋊C₄	—	86	1	no	—
261	C₄⋊C₄	—	86	1	no	—
262	C₄⋊C₄	—	86	1	no	—
263	C₄⋊C₄	—	86	1	no	—
264	C₄⋊C₄	—	86	1	no	—
265	C₄⋊C₄	—	86	1	no	—
266	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
267	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
270	C₂²⋊C₄	—	43	1	no	—
271	C₂²⋊C₄	—	43	1	no	—
272	C₂²⋊C₄	—	86	1	no	—
273	C₂²⋊C₄	—	86	1	no	—
274	C₂²⋊C₄	—	86	1	no	—
275	C₂²⋊C₄	—	86	1	no	—
276	C₂²⋊C₄	—	86	1	no	—
277	C₄²	—	43	1	no	—
278	C₄²	—	86	1	no	—
279	C₂³	—	43	1	no	—
280	C₂³	—	86	1	no	—
281	C₂³	D₄×D₄₃	1	1	yes	characteristic
282	Q₈	—	86	1	no	—
283	Q₈	—	86	1	no	—
284	Q₈	—	344	1	no	—
285	Q₈	—	172	2	no	—
286	D₄	—	43	4	no	—
287	D₄	—	86	1	no	—
288	D₄	—	86	1	no	—
289	D₄	—	172	2	no	—
290	D₄	—	172	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	43	1	no	—
300	C₂×C₄	—	43	1	no	—
301	C₂×C₄	—	43	1	no	—
302	C₂×C₄	—	86	1	no	—
303	C₂×C₄	—	86	1	no	—
304	C₂×C₄	—	86	1	no	—
305	C₂×C₄	—	86	1	no	—
306	C₂×C₄	—	86	1	no	—
307	C₂×C₄	—	344	1	no	—
308	C₂×C₄	—	172	2	no	—
309	C₂×C₄	—	172	2	no	—
310	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	86	1	no	—
314	C₈	—	86	1	no	—
315	C₈	#10621898	1	2	yes	complemented
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	43	2	no	—
319	C₂²	—	86	1	no	—
320	C₂²	—	86	1	no	—
321	C₂²	—	172	1	no	—
322	C₂²	—	172	1	no	—
323	C₂²	—	172	1	no	—
324	C₂²	SD₁₆×D₄₃	1	2	yes	—
325	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	43	2	no	—
332	C₄	—	86	1	no	—
333	C₄	—	86	1	no	—
334	C₄	—	172	1	no	—
335	C₄	—	172	1	no	—
336	C₄	—	172	1	no	—
337	C₄	C₂×#10621898	1	1	yes	characteristic
338	C₄	#13063373	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	86	2	no	—
341	C₂	—	344	1	no	—
342	C₂	#14583550	1	1	yes	characteristic
343	C₂	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
344	C₂	#14583705	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂×#106×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₂×#106	D₄₃
hypocenter	C₄₃	#3335
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #14583604	C₂	semidirect
3. #14583296	C₂	semidirect
4. #14583231	C₂	semidirect
5. #14583229	C₂	semidirect
6. #14583219	C₂	semidirect
7. #14583217	C₂	semidirect
8. #14583188	C₂	semidirect
9. #14583183	C₂	semidirect
10. #14583129	C₂	semidirect
11. #14583128	C₂	semidirect
12. #14582986	C₂	semidirect
13. C₈×C₂²⋊D₄₃	C₂	semidirect
14. #106×D₄₃	C₂	semidirect
15. C₂×#106×C₄₃	C₂	semidirect
120. C₂²⋊D₄₃	SD₁₆	semidirect
140. C₈×C₄₃	#41	semidirect
187. C₂×#106	D₄₃	semidirect
188. C₄₃	#3335	semidirect
315. C₈	#10621898	semidirect